When 3738, 5659 and 9501 are divided by the greatest possible number x, the remainder in each case is y. What is the sum of x and y ?

Options

A. 3738
B. 3673
C. 3783
D. 3637

Answer

To solve the problem, we need to find the greatest possible number \( x \) that divides the numbers 3738, 5659, and 9501 with the same remainder \( y \). This means that \( x \) must be a divisor of the differences between these numbers.

### Step 1: Calculate the differences between the numbers

1. **Difference between 5659 and 3738**:
\[ 5659 - 3738 = 1921 \]

2. **Difference between 9501 and 5659**:
\[ 9501 - 5659 = 3842 \]

3. **Difference between 9501 and 3738**:
\[ 9501 - 3738 = 5763 \]

### Step 2: Find the greatest common divisor (GCD) of the differences

Now we need to find the GCD of the differences calculated above: 1921, 3842, and 5763.

1. **GCD of 1921 and 3842**:
- The factors of 1921 are 1, 7, 11, 13, 19, 37, 53, 73, 91, 143, 221, 286, 1009, 1921.
- The factors of 3842 are 1, 2, 7, 11, 13, 19, 37, 53, 73, 91, 143, 221, 286, 1009, 1921, 3842.
- The common factors are 1, 7, 11, 13, 19, 37, 53, 73, 91, 143, 221, 286, 1009, 1921.
- The GCD is 1921.

2. **GCD of 1921 and 5763**:
- The factors of 5763 are 1, 3, 7, 13, 19, 27, 37, 63, 91, 143, 221, 273, 1009, 5763.
- The common factors with 1921 are 1, 7, 13, 19, 37, 91, 143, 221, 1009.
- The GCD is 91.

### Step 3: Identify \( x \) and \( y \)

The greatest possible number \( x \) is 91. To find \( y \), we can use any of the original numbers and find the remainder when divided by \( x \).

Using 3738:
\[ 3738 \div 91 = 41 \quad \text{(quotient)} \]
\[ 3738 - (41 \times 91) = 3738 - 3731 = 7 \]
So, \( y = 7 \).

### Step 4: Calculate the sum of \( x \) and \( y \)

Now, we can find the sum:
\[ x + y = 91 + 7 = 98 \]

### Final Answer
The sum of \( x \) and \( y \) is \( 98 \).

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